Multiple layered solutions of the nonlocal bistable equation

The nonlocal bistable equation is a model proposed recently to study materials whose constitutive relations among the variables that describe their states are nonlocal. It resembles the local bistable equation (the Allen–Cahn equation) in some way, but contains a much richer set of solutions. In this paper we consider two types of solutions. The first are the periodic solutions on a finite interval. These solutions are observed in materials like elastic crystals undergoing martensitic phase transitions and diblock copolymers at low temperatures. They are constructed by a variational method known as the Γ-limit technique. The second are solutions on the entire real line with transition layers, which are found by the formal matched asymptotics argument. We construct them to compare with the single layer heteroclinic and traveling wave solutions of the local bistable equation. The existence of multiple layered solutions depends on a unique nonlocal feature: the presence of two properly balanced competing effects of the constitutive relation, the oscillation inhibiting effect and the oscillation forcing effect, which coexist at two different length scales.